Pitch Plane Simulation of Aircraft Landing Gears Using

8/13/2019 Pitch Plane Simulation of Aircraft Landing Gears Using

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1998 International ADAMS User Conference {PAGE } Pitch Plane Simulation of Aircraft Landing Gearsusing ADAMS

Pitch Plane Simulation of Aircraft Landing Gears using ADAMS

H. Vinayak, Ph.D., Lead Analytical Development Engineer

J. J. Enright, Chief, Dynamics

Aircraft Wheels and Brakes

BFGoodrich AerospaceTroy, OH 45373

INTRODUCTION

Landing gears with four or more wheels and brakes

are common in modern commercial aircraft. Such

landing gears are subject to heavy loads and

dynamic forces. Much research has been done on

single axle (T-gear) systems (Figure 1a). Both

analytical and numerical/analog simulations of

such systems are available in literature (1-7).

However there are several research issuespertaining to Bogie-type landing gears with tandem

axes (Figure 1b) that have not been suitably

addressed by prior investigators. For instance, the

force coupling between fore and aft brakes and

effect of bogie pitch on the overall gear stability are

not well understood. This paper attempts to develop

an analytical model and present numerical

simulations using ADAMS to help understand

these phenomena.

Specific objectives of this paper are as follows: 1)

present pitch-plane ADAMS models of aircraftlanding gear, 2) present partial correlation with

experiments, 3) study effect of energy-sharing and

force-coupling between the different vibration

modes, 4) study relative stability of different squeal

vibration modes, i.e. out-of phase and in-phase

vibration of forward and aft brakes and 5) verify

fidelity of dynamometer simulations.

LANDING GEARS: PITCH PLANE MODEL

Pitch plane of a landing gear is defined as the

vertical plane in the direction of aircraft motion.

This plane is parallel to the bogie in a Bogie-Gearsystem.

The Bogie-gear model presented in this paper (Fig.

1b) consists of only pitch-plane degrees of freedom,

viz. low frequency (~10 Hz) gear walk and bogie

pitch, medium frequency (~ 30 Hz) chatter and

high frequency (~250 Hz) squeal modes. The model

also contains airplane vertical and fore-aft degrees

of freedom. These degrees of freedom are

comprehensively defined in reference(1).

Friction coupling between the rotors and stators is

modeled as a non-linear function of rub-velocity as

shown below. This function is obtained from sub-

scale and full-scale dynamometer tests.

{EMBED Equation.3 } (1,2,3)

Where P: Net Brake pressure (psi)A: Disk area (in2)

: Friction coefficient

r: Friction radius (in)

Vrub: Rub velocity (in/s)

: Angular velocity (rad/s)

N: Number of friction couples

Following approximation has been made in the

landing gear model presented in this paper.

{EMBED Equation.3 } (4)

A typical (Vrub) curve is shown in Figure 3. Tire-runway friction is also modeled as a non-linear

function of SLIP ratio between the tire and runway

as follows.




Where Fnormal: Vertical Contact Force (lb)

tire: Tire friction coefficient

rroll: Tire Roll radius (in)

tire: Tire Angular velocity (rad/s)

Vaxle: Axle Fore-aft velocity (in/s)

STABILITY ANALYSIS

The equations of motion for a T-Gear system are

described in detail in Reference 1. Stability of the

various vibration modes of the landing gear

depends on the system damping and the


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characteristics of the non-linear friction parameters

(ref. 6). For example, decoupled equations for the

squeal mode provides criteria for stability as shown

below.


Where Jsq: Squeal Inertia (i.e. Non-

rotating brake components)

Csq: Squeal damping

Ksq: Squeal Stiffness

Expanding brake in a Taylor series around an

operating point Vrubo(see Figure 3) we obtain


Substituting brake from equation (9) into equation

(1) and resulting expression for Tinto equation (8)we obtain equation (10).


It is clear from this equation that squeal DOF is

stable if

{EMBED Equation.3 }. (11)

Thus a brake will have unstable squeal mode if

equation (11) is violated. This can happen if the

friction material has a negative -Vrub slope(coefficient which decreases with increasing rub-velocity) that is greater than available positive

squeal damping in the system.

Similar stability criteria can be developed for the

other degrees of freedom of the landing gear.

ADAMS AIRCRAFT MODEL: DESCRIPTION

The T-Gear model consists of a total of 8 rigid

bodies coupled by spring-damper and non-linear

friction forces. The strut stiffness and effective gear

length (L) are obtained from modal analysis of the

gear supplied by the OEM (ref. 7). The brake and

wheel are modeled using measured rigid body

inertia and lumped stiffness parameters obtained

from finite element analysis of the components.

Torque, strut deflection (walk), squeal and aircraft

speed from a typical simulation of a landing stop

are plotted in Figures 4a,b,c and d respectively.

Notice the peaking of torque at the end of the stop

as speed decreases. This indicates a negative toque-

velocity slope resulting in the gear-walk instability.

The initial divergence of the walk deflection

corresponds to the net negative walk damping of

the system.

In the absence of tire slip, wheel chatter and brake

squeal, the rotational velocities of the rotor and

stator are given by the following expression.

{EMBED Equation.3 } (12 a,b)

Substituting Equation 12 in Equation 3, we obtain


Vaircraft decreases during a stop and { EMBED

Equation.3 } increases due to walk divergence.Towards the end of a stop, Vrub can become

negative during a back swing of the gear when {

EMBED Equation.3 } is negative. When this

happens, the brake torque as given by Equation (1),

reverses direction. This is manifested as sharp

spikes in the brake torque plotted in Figure 4a.

These torque-reversals limit walk vibration,

resulting in the linear convergence seen in Figure

4b. These sharp torque-reversal spikes excite

higher frequency squeal vibration that is quickly

attenuated by available positive squeal damping as

seen in Figure 4c.

ADAMS MODEL VALIDATION

L1011 landing gear was used for validation of the

model. ADAMS simulation is done for a complete

landing stop. Brake torque and walk degree-of-

freedom are plotted in Figure 5a and b. A half-scale

L1011 landing gear was also tested on a

dynamometer. Torque and strut deflection (walk)

obtained during one of the stops is plotted in Figure

5c and d. Compare these to ones obtained from

ADAMS simulation. It is clear that dynamic

response of the brake is very well predicted by the

ADAMS simulation.

VIBRATION MODE COUPLING

Due to the negative -Vrub characteristics of the

friction material, a certain amount of negative

damping exists in the system. In Figure 4, this

shows up as instability of walk mode. However, if


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by some means, one were to increase the positive

damping of this mode, one of the other modes

becomes unstable. This is illustrated clearly in

Figures 6 a,b and c. Walk, Squeal and Chatter

modes are plotted for three different damping

conditions in these figures.

Figure 6a corresponds to a case where walk is most

unstable resulting in this mode of vibration. As

explained in the previous section, walk couples

with squeal and chatter modes by providing higher

frequency excitations in the form of torque-

reversals.

In the second case, shown in Figure 6b, walk

damping was increased to make this mode stable.

However, this merely shifted the instability to the

higher frequency squeal mode. Now the torque-

reversals due to squeal vibration couples this modewith walk and chatter, exciting them to some

extent.

In the third case, shown in Figure 6c, squeal

damping was increased to stabilize this mode. Now

the chatter mode becomes unstable. Again, torque-

reversals due to chatter excites walk and squeal,

confirming the force coupling present between the

different vibration modes.

Dynamometer tests, using a fixture to represent the

walk mode, were conducted on a large commercial

aircraft brake with varying walk damping. Torque,pressure and aircraft speed obtained from these

tests are plotted in Figures 7a and b. The

oscillations in torque and pressure are proportional

to walk and squeal vibration respectively. Walk

vibration is more unstable with a damping of 2.8%.

As expected, walk vibration stabilizes when the

walk damping was increased to 5.2%. However,

this results in an increase in squeal vibration as

seen in Figure 5b. This confirms the predictions of

the ADAMS simulation discussed above.

RELATIVE STABILITY OF SQUEAL MODES

Two pitch plane squeal modes exist for a Bogie-

type landing gear. In the first mode, the fore and aft

brakes have an in-phase torsional oscillation. In the

second mode, the fore and aft brakes vibrate out-of-

phase. ADAMS prediction of fore and aft brake

squeal vibration in a Bogie-gear are plotted in

Figure 8. Initial squeal vibration is due to the

inherent squeal instability and is not initiated by

any particular disturbance. As seen in the figure,

this vibration is out-of-phase at 210 Hz. However,

later squeal vibration pulses are initiated by torque-

reversals that occur simultaneously in both the fore

and aft brakes. This results in an in-phase vibration

mode at a lower frequency of 137 Hz.

A conclusive assessment of the relative stability of

the two squeal modes can be made from Figure 9

which shows fore and aft brake vibration for a

typical stop with squeal instability. In this

simulation, both brakes had identical -Vrubfunction and the amount of positive viscous

damping in the in-phase and out-of-phase modes

were made equal. The vibration is initiated by a

runway bump which is applied to the two brakes

simultaneously. As expected, vibration starts out as

in-phase, but quickly changes to out-of phase. This

indicates that for this gear configuration, out-of-

phase squeal mode is less stable than in-phase

mode, possibly due to bogie pitch and other system

dynamics.

PITCH PLANE DYNAMOMETER SIMULATION

Extensive dynamometer tests of aircraft brakes are

conducted before qualification on aircraft.

Dynamics related tests require that the brake be

attached to mechanical simulators that have

impedance characteristics similar to aircraftlanding gear. Design and mathematical analysis of

such simulators are presented in detail in Reference

(1).

An ADAMS model of a bogie-gear simulator was

built to study the fidelity of dynamometer tests.

This model is shown in Figure 10. The fore-aft

motion of aircraft walk degree-of-freedom is

simulated as torsional motion of the simulator walk

beam. Simulator walk inertia, stiffness and

damping are adjusted to match those of aircraft.

Torque, walk and squeal motion for a typical stopwith walk vibration on the dynamometer are shown

in Figure 10a. Corresponding plots for the aircraft

simulation with identical operating conditions are

shown in Figure 10b. Note that amplitude of walk

is greater on the simulator than on the aircraft. This

is a result of small angle approximations made

while calculating the simulator parameters. Thus

dynamometer simulations are actually conservative,


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i.e. dynamometer tests would certainly identify

potential aircraft brake instabilities.

CONCLUSION

In summary, this paper presents results of ADAMS

modeling and analysis done at BFGoodrich,

Aerospace over the past few years. Basic ADAMS

simulation models have been built for T-gears,

Bogie-type gears and Dynamometer Simulators.

Four specific conclusions can be made from the

study presented in this paper. First, the ADAMS

models have been correlated with experimental

data. Second, the different modes of vibration are

coupled and stability of the landing gear is truly a

system parameter. Adding damping to a particular

unstable mode merely results in destabilizing or

reducing the stability margin of some other mode.

Third, aft brake in out-of-phase mode is the leaststable brake for the bogie-gear configuration

studied in this paper. Fourth, ADAMS simulation

have successfully verified fidelity of dynamometer

tests. In fact, it has been found that the pitch-plane

simulators are conservative. The basic ADAMS

models presented in this paper have been fully

parameterized and will be used as framework to

perform extensive design studies in the future.

ACKNOWLEDGEMENT

Thanks to L.D. Bok, Vice President. Engineering

and D.W. Moser, Manager, Technologies forproviding a conducive environment for basic

research in a manufacturing setting that made the

work presented in this paper possible.

REFERENCES

1. J. J. Enright 1986 Society of AutomotiveEngineers, No. 851937, laboratory Simulation

of Landing Gear Pitch-Plane Dynamics.

2. J. T. Gordon 1997 ASME Design andTechnical Conferences, DETC97/VIB-4163,

A perturbation Analysis of Nonlinear SquealVibrations in Aircraft Braking Systems.

3. S.Y. Liu, J.T. Gordon, M.A. Ozbek 1996AIAA Conference Proceedings, AIAA-96-

1251-CP, A Nonlinear Model For Aircraft

Brake Squeal Analysis, Part I: Model

Description and Solution Methodology.

4. J.T. Gordon, S.Y. Liu, M.A. Ozbek 1996AIAA Conference Proceedings, AIAA-96-

1251-CP, pp. 406-416 A Nonlinear Model For

Aircraft Brake Squeal Analysis, Part II:

Stability Analysis and parametric Studies.

5. M.H. Travis 1995 ASME Design EngineeringTechnical Conference, DE-Vol. 84-1,pp.1209-1216, pp. 417-426, Nonlinear

Transient Analysis of Aircraft landing Gear

Brake Whirl and Squeal.

6. R. H. Black 1995 ASME Design EngineeringTechnical Conference, DE-Vol. 84-1, pp.

1241-1245, Self excited Multi-Mode

Vibrations of Aircraft brakes with Nonlinear

Negative Damping.

7. C. F. Chang, 1995 ASME Design EngineeringTechnical Conference, DE-Vol. 84-1, pp.1217-1227, The Dynamic finite Element

Modeling of Aircraft Landing Gear System.


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a) T-Gear System

b) Bogie-Gear System

Figure 1.ADAMS Simulation Models


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FIGURE 2. SCHEMATIC OF A LANDING GEAR SYSTEM

Figure 3. Typical -VrubPlot

Vrub

brak

o

Vrubo

etc,T,VrubV

1


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Figure 4. Torque, Walk, Squeal, A/C Speed and Wheel Speed for a typical stop.

a) L1011 ADAMS: Torque c) L1011 Dynamometer: Torque

b) L1011 ADAMS: Walk d) L1011 Dynamometer: Walk

FIGURE 5. COMPARISON OF ADAMS PREDICTION TO DYNAMOMETER TESTS

a)

b)

c

d


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Figure 6. Energy sharing between different modes of vibration

a) Walk Instability

b) Squeal Instability

c) Chatter Instability

Tor ue

Walk

Squeal

Chatter

Tor ue

Walk

Squeal

Chatter

Tor ue

Walk

Squeal

Chatter


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a) Aircraft walk damping = 2.8% b) Aircraft walk damping = 5.2%

FIGURE 7. DYNAMOMETER TESTING: ENERGY SHARING BETWEEN MODES

Figure 8. Relative stability of in-phase and out-of-phase squeal modes


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Figure 9. ADAMS T-Gear Simulation: Squeal initiated by runway bump. Indicates: a) out-of

phase mode is more unstable than in-phase mode and b) aft-brake is more unstable than fore-

brake.


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Figure 10. ADAMS model of landing Gear Simulator

a) Dynamometer b) Aircraft

Figure 11a. ADAMS Simulation

WALK INERTIA

WALK DAMPER

TORSIONAL

WALK STIFFNESS

Torque

Walk

Chatter

Squeal

Torque

Walk

Chatter

Squeal


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